Ramified Characters of Idèle Groups of One-Class Quadratic Fields

The necessary and sufficient conditions for the triviality of the ramified characters of the idèle group $A_d^\times $ of a one-class quadratic field $\mathbb Q(\sqrt d)$ on its multiplicative group $\mathbb Q^\times(\sqrt d)$ are established; i.e., the characters of the group $A_d^\times/\mathbb Q^\times(\sqrt d)$ are completely described.